This document summarizes research on modeling the spread of plant pathogens in complex networks. It discusses simulating disease spread in local, small-world, random and scale-free networks. Examples show epidemic development varies in these networks. The probability of persistence and transmission determines epidemic threshold. Final epidemic size depends on the starting node's links. Spatially-explicit modeling and implications for landscape management are also covered.

1. Modelling the spread of
Phytophthora ramorum
in complex directed networks
Marco Pautasso,
Division of Biology,
Imperial College London,
Wye Campus, Kent, UK
Wye, 14 Jul 2007

2. Sudden Oak Death
from Desprez-Loustau et multi al. (in press) Trends in Ecology & Evolution

3. Trace-forwards and positive detections across the USA, July 2004
Trace forward/back zipcode
Positive (Phytophthora ramorum) site
Hold released
Source: United States Department of Agriculture,
Animal and Plant Health Inspection Service, Plant Protection and Quarantine

4. Simulation of disease spread in four basic
types of directed networks of small size
local small-
world SIS-model
N nodes = 100
constant n of links
directed networks
probability of infection
for the node x at time
random scale- t+1 = Σ px,y iy where
free px,y is the probability
of connection between
node x and y, and iy is
the infection status of
the node y at time t
from: Pautasso & Jeger (in press) Ecological Complexity

5. Examples of epidemic development in four kinds
of directed networks (at threshold conditions)
sum probability of infection across all nodes
1.2 40 1.2 25
35
% nodes with probability of infection > 0.01
1.0 1.0
20
small-world network nr 4;
30
0.8 0.8
25
starting node = nr 14 15
0.6 20 0.6
10
15
0.4 0.4
local network nr 6; 10
5
starting node = nr 100
0.2 0.2
5
0.0 0 0.0 0
1 51 101 151 201 1 26 51 76
1.6 iteration 60
1.2 iteration 80
1.4
1.0
scale-free network nr 2; 70
starting node = nr 11
50
1.2 60
40 0.8
1.0 50
0.8 30 0.6 40
0.6
random network nr 8; 30
0.4
starting node = nr 80 20 0.4
20
10 0.2
0.2 10
0.0 0 0.0 0
1 26 51 76 1 26 51 76
from: Pautasso & Jeger (in press) Ecological Complexity

6. Linear epidemic threshold on a plot of
p(persistence) f p(transmission)
1.00
local
epidemic
develops small-world
probability of persistence
0.75 random
scale-free
0.50
0.25
no
epidemic
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
probability of transmission
from: Pautasso & Jeger (in press) Ecological Complexity

7. 2.0 2.5
infection probability across all nodes) local 2.0
small-world
1.5
Final size of epidemic (sum of
1.5
1.0
1.0
0.5
0.5
0.0 0.0
0 25 50 75 100 0 25 50 75 100
3.0 6.0
2.5
random 5.0 scale-free
2.0 4.0
1.5 3.0
1.0 2.0
0.5 1.0
0.0 0.0
0 25 50 75 100 0 25 50 75 100
Starting node of epidemic
from: Pautasso & Jeger (in review) Journal of Theoretical Biology

8. Marked variations in epidemic final size at threshold conditions
depend on the number of links of the starting node
sum at equilibrium of probability
2.0 3.0
local 2.5 small-world
of infection across all nodes
1.5
2.0
1.0 1.5
1.0
0.5 0.5
0.0
0.0
0 2 4 6 8
0 1 2 3 4 5 6
3.0 6.0
random 5.0
scale-free
2.5
2.0 4.0
1.5 3.0
1.0 2.0
0.5 1.0
0.0 0.0
0 2 4 6 8 10 12 0 20 40 60 80 100
n of links from starting node n of links from starting node
from: Pautasso & Jeger (in review) Journal of Theoretical Biology

9. Spatially-explicit
modelling framework
Climate Long-distance trade
suitability
Local Trade
Heathland
Woodland

10. Network epidemiology
Nature's guide for mentors
number of passengers per day
from: Hufnagel, Brockmann & Geisel (2004) PNAS

11. Epidemiology is just one of the
many applications of network theory
NATURAL
Network pictures from:
Newman (2003) SIAM Review
food webs
cell
metabolism
neural Food web of Little Rock
networks Lake, Wisconsin, US
ant nests sexual
partnerships
DISEASE
SPREAD
family
innovation networks
Internet flows co-authorship HIV
structure railway urban road nets spread
electrical networks networks network
power grids telephone calls
WWW
computing airport Internet E-mail
committees
grids networks software maps patterns
TECHNOLOGICAL SOCIAL
Modified from: Jeger, Pautasso, Holdenrieder & Shaw (2007) New Phytologist

12. Acknowledgements
Peter Weisberg, Chris Gilligan, Univ.
Univ. of Nevada, of Cambridge
Reno, US Mike Jeger,
Ottmar Imperial College, Mike Shaw,
Holdenrieder, Wye Univ. of
ETHZ, CH Reading
Kevin
Gaston,
Mike Univ. of
Sheffield Emanuele Della
McKinney, Katrin Valle, Politecnico di
Univ. of Boehning Milano, Italy
Tennessee, -Gaese,
US Univ. Mainz, Germany

13. References
Dehnen-Schmutz K, Holdenrieder O, Jeger MJ & Pautasso M (2010) Structural change in the international horticultural industry: some implications
for plant health. Scientia Horticulturae 125: 1-15
Harwood TD, Xu XM, Pautasso M, Jeger MJ & Shaw M (2009) Epidemiological risk assessment using linked network and grid based modelling:
Phytophthora ramorum and P. kernoviae in the UK. Ecological Modelling 220: 3353-3361
Jeger MJ & Pautasso M (2008) Comparative epidemiology of zoosporic plant pathogens. European Journal of Plant Pathology 122: 111-126
Jeger MJ, Pautasso M, Holdenrieder O & Shaw MW (2007) Modelling disease spread and control in networks: implications for plant sciences. New
Phytologist 174: 179-197
Lonsdale D, Pautasso M & Holdenrieder O (2008) Wood-decaying fungi in the forest: conservation needs and management options. European
Journal of Forest Research 127: 1-22
MacLeod A, Pautasso M, Jeger MJ & Haines-Young R (2010) Evolution of the international regulation of plant pests and challenges for future plant
health. Food Security 2: 49-70
Moslonka-Lefebvre M, Pautasso M & Jeger MJ (2009) Disease spread in small-size directed networks: epidemic threshold, correlation between
links to and from nodes, and clustering. J Theor Biol 260: 402-411
Moslonka-Lefebvre M, Finley A, Dorigatti I, Dehnen-Schmutz K, Harwood T, Jeger MJ, Xu XM, Holdenrieder O & Pautasso M (2011) Networks in
plant epidemiology: from genes to landscapes, countries and continents. Phytopathology 101: 392-403
Pautasso M (2009) Geographical genetics and the conservation of forest trees. Perspectives in Plant Ecology, Systematics & Evolution 11: 157-189
Pautasso M (2010) Worsening file-drawer problem in the abstracts of natural, medical and social science databases. Scientometrics 85: 193-202
Pautasso M & Jeger MJ (2008) Epidemic threshold and network structure: the interplay of probability of transmission and of persistence in directed
networks. Ecological Complexity 5: 1-8
Pautasso M et al (2010) Plant health and global change – some implications for landscape management. Biological Reviews 85: 729-755
Pautasso M, Moslonka-Lefebvre M & Jeger MJ (2010) The number of links to and from the starting node as a predictor of epidemic size in small-
size directed networks. Ecological Complexity 7: 424-432
Pautasso M, Xu XM, Jeger MJ, Harwood T, Moslonka-Lefebvre M & Pellis L (2010) Disease spread in small-size directed trade networks: the role of
hierarchical categories. Journal of Applied Ecology 47: 1300-1309
Xu XM, Harwood TD, Pautasso M & Jeger MJ (2009) Spatio-temporal analysis of an invasive plant pathogen (Phytophthora ramorum) in England
and Wales. Ecography 32: 504-516